In general relativity, an inertial frame can only be established in a small
region of spacetime, and a locally inertial frame is mathematically represented
by a tetrad field in gravity. The tetrad field is not unique due to the freedom
to perform a local Lorentz transformation in an inertial frame, and there
exists freedom to choose the locally inertial frame at each spacetime. The
local Lorentz transformations are known as non-Abelian gauge transformations
for the tetrad field, and to fix the gauge freedom, corresponding to the
Lorentz gauge ∂μAμ​=0 and Coulomb gauge
∂iAi​=0 in electrodynamics, the Lorentz gauge and Coulomb
gauge for the tetrad field are proposed in the present work. Moreover,
properties of the Lorentz gauge and Coulomb gauge for tetrad field are
discussed, which show the similarities to those in electromagnetic field.Comment: 4 pages, no figure, comments are welcome